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8/7/2019 Prabhat Ppt
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PRESENTATION
on
Axiomatizations
for probabilistic finite-state behavioursBy:Prabhat kumar(A3/64)
Under the guidence of:Mr. ArpitDepartment of CSE
IERT Allahabad
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CONTENTS
Introduction
AxiomatizationProbabilistic automata
Probabilistic models
Axiomatic systemBehavioral equivalences
Conclusion
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INTRODUCTION
The reason why we are interested in studying a model which expresses both nondeterministic and probabilistic behavior, and an equivalence sensitive to divergency, is that one of the long-term goals of this line of research is to
develop a theory which will allow us to reason about probabilistic algorithms used in distributed computing. In that domain it is important to ensure that an algorithm will work under any scheduler, and under other unknown oruncontrollable factors.
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Axiomatization
Axiomatization is the formulation of a system
ofstatements(i.e.axioms)thatrelateanumber
of primitive terms in order that
a consistent body of propositions may be
derived deductively from these statements.
Thereafter, theproofofanypropositionshouldbe,inprinciple,traceablebacktotheseaxioms.
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PROBABILISTIC AUTOMATA
Basically, a probabilistic automaton is just an ordinary automaton (also
called labeled transition system or state machine) with the only difference
thatthetargetofatransitionisaprobabilisticchoiceoverseveralnextstates.
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PROBABILISTIC MODELS
In 1995 van Glabbeek et al. classify probabilisticmodels into following three models:-
Reactive
Generative
Stratified
After that Segala pointed out that neitherreactive nor generative nor stratified models
capture real nondeterminism, an essentialnotion for modeling scheduling freedom. Hethen introduced two models:-
Probabilistic automata (PA)
Simple probabilistic automata (SPA)
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1) Inreactivemodels,eachlabeledtransitionisassociatedwithaprobability,andforeachstatethesumoftheprobabilitieswiththesame
labelis1.
2) Ingenerativemodelsforeachstatethesumoftheprobabilitiesofalltheoutgoingtransitionsis1.
3) Stratifiedmodelshavemorestructureandforeachstateeitherthereisexactlyoneoutgoinglabeledtransitionorthereareonly
unlabeledtransitionsandthesumoftheirprobabilitiesis1.
4) SPAis asimplifiedversionofPAcalledsimpleprobabilisticautomatawhicharelikeordinaryautomataexceptthatalabeled
transitionleadstoaprobabilisticdistributionoverasetofstatesinsteadofasinglestate.
5) InProbabilisticautomata(PA),bothprobabilityandnondeterminismaretakenintoaccount.Probabilisticchoiceisexpressedbythenotionoftransition,which,inPA,leadstoaprobabilisticdistributionoverpairs(action,state)anddeadlock.Nondeterministic
choice,ontheotherhand,isexpressedbythepossibilityofchoosingdifferenttransitions
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AXIOMATIC SYSTEM
Amathematicaltheoryconsistsofanaxiomaticsystemandallitsderivedtheorems.
Anaxiomatic systemthatiscompletelydescribedisaspecial kindofformal system; usuallythough theeffort towards complete formalisation brings diminishing returns in certainty, and a lack of readability
forhumans.Thereforediscussiono A axiomatic system is any set of axioms from which some or all axioms can be used in conjunction
tologicallyderivetheorems.
f axiomatic systems is normally only semi-formal. A formal theory typically means an axiomaticsystem,forexampleformulatedwithinmodeltheory.
Aformalproofisacompleterenditionofamathematicalproofwithinaformalsystem.
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BEHAVIORALEQUIVALENCES
Strong bisimulationStrong probabilistic bisimulation
Weak bisimulation
Weak probabilistic bisimulationDivergency-sensitive equivalence
Observational equivalence
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CONCLUSION
Here we have proposed a probabilistic process calculus which corresponds tosegalas probabilistic automata We have presented strong bisimulation, strongprobabilisticbisimulation.Wehaveaxiomatized divergency-sensitive equivalence
and observational equivalenceonlyforguardedexpressions.Weconjecturethatthe two behavioral equivalences are undecidable and therefore not finitelyaxiomatizable.In thefuture itmight beintresting toseehow to refineour processalgebratoallowforparallelcomposition.
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